Clustering and percolation of point processes

TL;DR

This paper investigates phase transitions in percolation models on point processes, demonstrating how clustering properties influence percolation, with examples including determinantal, perturbed lattices, and negatively associated processes.

Contribution

It establishes conditions under which non-trivial phase transitions occur in coverage models based on point process clustering properties, extending understanding of percolation in complex spatial structures.

Findings

Point processes with less clustering than Poisson exhibit phase transitions.

Existence of phase transition in ech simplicial complex percolation is confirmed.

A Cox process with high clustering percolates at arbitrarily small radii.

Abstract

We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson point process exhibit non-trivial phase transition in the percolation of some coverage models based on level-sets of additive functionals of the point process. Examples of such point processes are determinantal point processes, some perturbed lattices, and more generally, negatively associated point processes. Examples of such coverage models are $k$-coverage in the Boolean model (coverage by at least $k$ grains) and SINR-coverage (coverage if the signal-to-interference-and-noise ratio is large). In particular, we answer in affirmative the hypothesis of existence of phase transition in the percolation of $k$-faces in the \v{C}ech simplicial complex (called also clique percolation) on point processes which cluster less than the Poisson process. We also construct a Cox point process, which is "more clustered" than the Poisson point process and whose Boolean model percolates for arbitrarily small radius. This shows that clustering (at least, as detected by our specific tools) does not always "worsen" percolation, as well as that upper-bounding this clustering by a Poisson process is a consequential assumption for the phase transition to hold.